When the Italian fashion designer Miuccia Prada is in especially high spirits, she leaves her office on a slide. The entrance to it, a stainless-steel funnel that looks like an airlock leading into a spaceship, emerges from the middle of the polished concrete floor of her minimalist office, which is on the third floor of the Prada Group's headquarters in Milan. After burrowing through brickwork, the slide suddenly appears in mid-air, before coiling and dipping into the courtyard below.

As well as providing a rapid means of escape for Mrs Prada – as she is always called by her staff – the slide is also a work of art, by the German artist Carsten Höller. In 2006, Höller famously installed five similar slides in Tate Modern, where they were a huge hit. "We have reached a great changing point in art," Prada tells me. "What interests me most is when a work of art is no longer just an object, but also touches reality and life."

A former mime student and member of the Communist Party, with a PhD in political science, in the Seventies Prada reluctantly took over the family business, founded by her paternal grandfather Mario in 1913. Over the following decades, she turned it from a sleepy store specialising in leather luggage for wealthy Italians into a fashion superpower, selling clothes and accessories in nearly 80 countries around the world. In the 2007 fiscal year, the group's turnover reached £1.2 billion.

What is her secret? One answer is that Prada has consistently pursued a very particular and unusual aesthetic. She often makes clothes that are not obviously sexy: "I once tried to make lace – which has been a great obsession of women – unsexy. And I achieved it," she tells me with pride. She was one of the first designers to become interested in military uniforms; one of the first to look to vintage dresses for inspiration. Her much-imitated backpack made from black industrial nylon and trimmed with leather became the must-have accessory of the late Eighties and Nineties (it still sells well today).

Sometimes, her creations look frumpy, even ugly, but they all manage the tricky feat of being widely desirable yet conferring on their owners a sense that they are discerning. It is often said that Prada makes clothes not for women's bodies but for their brains (she never designs dresses that flaunt much flesh). Her creations are covetable, but different – a cut above the more garish designs of other Italian houses such as Versace.

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Increasingly, however, Prada's interests have broadened away from the confines of fashion design to the subject of our conversation today: her passion for contemporary art. In the early Nineties, Prada and her husband, Patrizio Bertelli, a flamboyant Tuscan who is the group's chief executive, began to collect post-war and contemporary art. Today, visitors to their apartment in Milan, the same apartment in which Prada was born in 1949, can see pieces by important artists of the Sixties, including Lucio Fontana and Blinky Palermo.

But it is the Prada Foundation, their not-for-profit organisation devoted to contemporary art, that has really earned them the respect of the international art world. Established in 1995, the foundation has realised ambitious projects with some of the most exciting names in contemporary art, including several celebrated Brits (Anish Kapoor, Steve McQueen, Marc Quinn, and Sam Taylor-Wood). One of the most popular installations was Höller's Upside Down Mushroom Room, in which gigantic, topsy-turvy toadstools slowly revolved from the ceiling. It was visited by 13,000 people in just a month.

The foundation's most recent project, The Double Club, a nightspot-cum-artwork dreamed up by Höller that presents a beguiling fusion of Western and Congolese cultures, occupies a warehouse in north London. To coincide with the Venice Biennale in June, the foundation will present a retrospective of work by the neglected Californian pop artist John Wesley on the island of San Giorgio Maggiore. "We will have more than 150 beautiful paintings," Prada says, sipping a cup of Earl Grey. "His work is very complex: surreal, pop, influenced by comics. He still seems contemporary, yet he is not really known. He's very shy, and that's part of the reason. Also he's subtle – his work was not obviously pop when pop was the big thing."

When did she and her husband become interested in contemporary art? "Our interest came about almost by chance," Prada says. "We had some friends who were sculptors, and when they saw one of our buildings in an old industrial space, they said, 'This is perfect for sculpture. You should do exhibitions.' And my husband said, 'That's a good idea'."

In the years that followed, they studied hard. "The only way to do something in depth is to work hard," she says. "We started meeting artists, visiting them. It was an intense training. We also started buying, but not with the idea of collecting, because I don't like the idea of being a collector at all." She laughs. "For me, art is about learning and about living with people. It's alive. Collecting is a little bit dead."

Behind her, propped against the wall, are two stylish prints by the Italian artist Francesco Vezzoli. They can't have been cheap, I suggest. As a former Communist, is she not sometimes horrified by the prices commanded by successful artists? "Money and art: it's a difficult question," she says. "But art has always been about very rich people – think of the great popes and princes of the Renaissance." She pauses. "But even if you are rich, you still have the right to your ideas. When I was involved with politics, I never wore jeans. On protests, I would dress in [Yves] Saint Laurent. So I have always lived with this kind of contradiction."

Art and fashion have frequently overlapped: Salvador Dalí collaborated with the Italian designer Elsa Schiaparelli, for instance, to produce the surrealist "lobster dress" modelled by Wallis Simpson shortly before she married the Duke of Windsor. But, for years, Prada has resisted mingling her interest in contemporary art with designing clothes. "Until very recently," she says, "I wanted to keep the two fields completely separate. I didn't want my work to be influenced by art in any way."

Why not? "I don't know. Probably because I used to think that art was high [status] and fashion was low, and that one was more moral than the other. This is a complex from the Sixties, because I was born in the protest movements of '68. But art is so important for me, and now I want to stop making this separation."

So is fashion design art? "I do commercial work," she says, firmly. "If I was only creative, I would become an artist. A designer can be very creative, but art is something that stands by itself, and fashion is something you sell."

When I ask her to describe her taste in art, Prada talks about qualities that I suspect she values in her own character as well as her quirky designs. "My husband has an incredible eye for what is beautiful, but in a traditional way. I tend to buy art that is not really beautiful, but which intrigues me because it is new. Good artists are always trying to go against something or trying to be innovative in a way that is not obvious."

"Miuccia's taste in art is more experimental than Patrizio's – which is also her role in Prada," says Germano Celant, the director of the Prada Foundation, which will soon occupy a new home, designed by the Dutch architect Rem Koolhaas, in a sprawling industrial site in south Milan. "Miuccia has learnt from art that, in order to do new things, you need to go against your confirmed style. She likes people who are questioning."

Does she see herself this way, I wonder? "Yes," she says. "I doubt everything. I always resist things that are obvious, even though what usually sells is the most obvious stuff. That puts me in an uncomfortable position."

Does such contrariness take its toll? "It is very tiring. I have to do eight shows a year and convince people all over the world to buy what we are doing. It's a lot. It's very hard. You can't do more than two bad shows in a row, otherwise you are dead. You have to keep the level, the excitement. But being creative every single day is really difficult. Sometimes I would prefer to do anything else in the company instead, such as taking care of the economics, just as a relief."

During our conversation, Prada comes across as both bashful and assertive; a visionary afflicted with self-doubt. I put it to her that her questing, quixotic approach to fashion and art is the secret behind her success. "It's my nature," she says. "I always want to be different, as a way to progress. At the beginning, I wanted to make a soft bag out of stiff leather. I wanted to make rich materials look poor, and poor materials look rich. Always there was something disturbing. In the end, that's probably why people like Prada."

As she talks, my eye is drawn again to the reflective funnel of Höller's slide. Can I have a go, I wonder? Prada looks delighted. "Of course," she says, and scuttles across the room to lift off the glass hatch. I say goodbye and climb on board. "Don't forget to scream!" she says with a wicked laugh, before giving me a push. Her laughter is still ringing in my ears when I land in the courtyard.

Carsten Höller's 'Double Club', London EC1 (020 7837 2222), is taking bookings until July. Details: thedoubleclub.co.uk 'John Wesley' is at the Fondazione Giorgio Cini on the island of San Giorgio Maggiore, Venice from June to Oct. Details: fondazioneprada.org.